# A topological splitting theorem for Poincar\'e duality groups and high dimensional manifolds

Niblo, Graham and Kar, Aditi (2013) A topological splitting theorem for Poincar\'e duality groups and high dimensional manifolds Geometry & Topology, 17, pp. 2203-2221.

## Description/Abstract

Waldhausen's celebrated torus theorem plays a central role in the classification of topological 3-manifolds. It also led to a number of algebraic splitting theorems for discrete groups including Kropholler's algebraic torus theorem for Poincar\'e duality groups and to the algebraic annulus theorems of Dunwoody/Sageev and Scott/Swarup. Here, in the same spirit, we offer topological and algebraic decomposition theorems in the context of high dimensional aspherical manifolds, providing an algebraic splitting theorem for Poincar\'e duality groups and exploiting Cappell's splitting theory to extract the required topological splittings. As a result we show that for a wide class of manifold pairs $N,M$ with $\dim(M)=\dim(N)+1$, every, $\pi_1$-injective map f$N\rightarrow M$ factorises up to homotopy as a finite cover of an embedding. As an application of this we show that under certain circumstances the vanishing of the first Betti number for $M$ is an obstruction to the existence of such maps.

Item Type: Article
Digital Object Identifier (DOI): doi:10.2140/gt.2013.17.2203
ISSNs: 1465-3060 (print)
Related URLs:
Keywords: Torus theorem, Poincaré duality group, Bass-Serre theory, Kazhdan's property (T), Borel conjecture, surgery, Cappell's splitting theorem, embeddings, rigidity, geometric group theory, quaternionic hyperbolic manifolds
Subjects:
Organisations: Pure Mathematics
ePrint ID: 161381
Date :
Date Event
28 July 2010Submitted
26 April 2013Accepted/In Press
26 July 2013Published
Date Deposited: 28 Jul 2010 18:59